DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Computable and partial computable functions

Definition

Fix effective encodings of sets $X$ and $Y$ .

A function $f : D \to Y, D \subseteq X,$ is partial computable when some deterministic Turing machine computes it in the sense of Partial functions computed by a machine under fixed encodings.

If $D = X$ , so the machine halts with the correct output on every input in $X$ , then $f$ is computable or total computable.

Remarks

"Computable" is the total case of "partial computable"; the distinction is whether divergence is allowed outside a proper domain.
The encoded machine works on binary words, but the represented function lives on the underlying sets $X$ and $Y$ through the chosen encodings.

Depends on

Partial functions computed by a machine under fixed encodings

Used by

The busy beaver function is not computable Corollary
Completeness at an arithmetical level Definition
Computable many-one reductions between languages Definition
Computably enumerable sets and languages Definition
Decidable and recognizable languages Definition
Description machines and plain Kolmogorov complexity Definition
Effectively open sets in Cantor space Definition
Limit-computable functions Definition
Domains and ranges of partial computable functions are computably enumerable Theorem
Every computably enumerable set is the domain of a partial computable function Theorem
Gap and union theorems for abstract complexity bounds Theorem
Infinite computably enumerable sets have computable injective enumerations Theorem
Kolmogorov complexity is not computable Theorem
Sigma₁⁰ sets are exactly the computably enumerable sets Theorem
The busy beaver function eventually dominates every total computable function Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
John Watrous, Introduction to the Theory of Computing, Lecture 18: Further discussion of computability (standard reference, not scraped)